Theorem lpss

Description: The limit points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007)

	Ref	Expression
Hypothesis	lpfval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	lpss	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	⊢ 𝑋 = ∪ 𝐽
2	1	lpsscls	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )
3	1	clsss3	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )
4	2 3	sstrd	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )